To find the indefinite integral of $\frac{d}{dx} (x^8 + 4x^2 + 4)$ with respect to $x$, we first simplify the expression inside the integral.

Step 1: Differentiate the Inner Function

Differentiate $x^8 + 4x^2 + 4$ with respect to $x$:

$\frac{d}{dx}(x^8 + 4x^2 + 4) = 8x^7 + 8x$

Step 2: Integrate the Result

Now, we integrate the resulting expression $8x^7 + 8x$ with respect to $x$:

$\int (8x^7 + 8x) \, dx$

Step 3: Apply the Power Rule for Integration

Using the power rule for integration, we integrate each term separately:

1. $\int 8x^7 \, dx = 8 \cdot \frac{x^8}{8} = x^8$
2. $\int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2$

Combining these results:

$x^8 + 4x^2 + C$

where $C$ is the constant of integration.

Final Result

The indefinite integral is:

$\int \frac{d}{dx} \left( x^8 + 4x^2 + 4 \right) \, dx = x^8 + 4x^2 + C$

This means that integrating the derivative of a function essentially gives us back the original function, up to an added constant of integration. If you have any further questions or need additional details, feel free to ask!

Here are some related questions to explore this topic further:

1. What is the significance of the constant of integration?
2. How do we differentiate and integrate polynomial functions?
3. What are the applications of indefinite integrals in physics?
4. How does the Fundamental Theorem of Calculus relate to this problem?
5. What are some common techniques for integrating more complex functions?
6. How do definite and indefinite integrals differ?
7. Can all functions be integrated indefinitely?
8. How does integration relate to finding the area under a curve?

Tip: When differentiating and integrating functions, always double-check your work by differentiating your integral to ensure it matches the original function.